Best constants for Lipschitz embeddings of metric spaces into c₀

N. J. Kalton; G. Lancien

Displaying similar documents to “Best constants for Lipschitz embeddings of metric spaces into c₀”

Biseparating maps on generalized Lipschitz spaces

Denny H. Leung (2010)

Studia Mathematica

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Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F-valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are...

Bi-Lipschitz embeddings of hyperspaces of compact sets

Jeremy T. Tyson (2005)

Fundamenta Mathematicae

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We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in ℓ². If X is a countable compact metric space containing at most n nonisolated points, there is a Lipschitz embedding of K(X) in $ℝ^{n + 1}$ ; in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. By way of contrast, the hyperspace K([0,1])...

Products of Lipschitz-free spaces and applications

Pedro Levit Kaufmann (2015)

Studia Mathematica

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We show that, given a Banach space X, the Lipschitz-free space over X, denoted by ℱ(X), is isomorphic to ${(\sum_{n = 1}^{\infty} ℱ (X))}_{ℓ ₁}$ . Some applications are presented, including a nonlinear version of Pełczyński’s decomposition method for Lipschitz-free spaces and the identification up to isomorphism between ℱ(ℝⁿ) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into ℝⁿ and which contains a subset that is Lipschitz equivalent to the unit ball of ℝⁿ. We also show...

A new proof of Fréchet differentiability of Lipschitz functions

Joram Lindenstrauss, David Preiss (2000)

Journal of the European Mathematical Society

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We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on $ℓ_{2}$ (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the $w^{*}$ closure of the set of its points of Gâteaux differentiability is norm separable.

On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček (1997)

Commentationes Mathematicae Universitatis Carolinae

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We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux differentiable bump function. In particular, our result implies the following theorem: If $d$ is a distance function determined by a closed subset $A$ of a Banach space $X$ with a uniformly Gâteaux differentiable norm, then the set of points of $X ∖ A$ at which $d$ is not Gâteaux differentiable is not only a first category set, but...

Two applications of smoothness in C(K) spaces

Matías Raja (2014)

Studia Mathematica

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A simple observation about embeddings of smooth Banach spaces into C(K) spaces allows us to construct a parametrization of the separable Banach spaces using closed subsets of the interval [0,1]. The same idea is applied to the study of the isometric embedding of $ℓ_{p}$ spaces into certain C(K) spaces with the additional condition that the functions of the image must be Lipschitz with respect to a fixed finer metric on K. The feasibility of that kind of embeddings is related to Szlenk indices. ...

Invariance of regularity conditions under definable, locally Lipschitz, weakly bi-Lipschitz mappings

Małgorzata Czapla (2010)

Annales Polonici Mathematici

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We describe the notion of a weakly Lipschitz mapping on a $C^{q}$ stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the Verdier condition.

A Lipschitz function which is $C^{\infty}$ on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

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We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is $C^{\infty}$ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...

Note on bi-Lipschitz embeddings into normed spaces

Jiří Matoušek (1992)

Commentationes Mathematicae Universitatis Carolinae

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Let $(X, d)$ , $(Y, ρ)$ be metric spaces and $f : X \to Y$ an injective mapping. We put ${∥ f ∥}_{Lip} = sup {ρ (f (x), f (y)) / d (x, y); x, y \in X, x \neq y}$ , and $dist (f) = {∥ f ∥}_{Lip} . {∥ f^{- 1} ∥}_{Lip}$ (the of the mapping $f$ ). We investigate the minimum dimension $N$ such that every $n$ -point metric space can be embedded into the space $ℓ_{\infty}^{N}$ with a prescribed distortion $D$ . We obtain that this is possible for $N \geq C {(log n)}^{2} n^{3 / D}$ , where $C$ is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into $ℓ_{p}^{N}$ are obtained by a similar method. ...

Double sine series with nonnegative coefficients and Lipschitz classes

Vanda Fülöp (2006)

Colloquium Mathematicae

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Denote by $f_{s s} (x, y)$ the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that $f_{s s}$ belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

Singular points of order $k$ of Clarke regular and arbitrary functions

Luděk Zajíček (2012)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a separable Banach space and $f$ a locally Lipschitz real function on $X$ . For $k \in ℕ$ , let $Σ_{k} (f)$ be the set of points $x \in X$ , at which the Clarke subdifferential $\partial^{C} f (x)$ is at least $k$ -dimensional. It is well-known that if $f$ is convex or semiconvex (semiconcave), then $Σ_{k} (f)$ can be covered by countably many Lipschitz surfaces of codimension $k$ . We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe,...

Lipschitz and uniform embeddings into $ℓ_{\infty}$

N. J. Kalton (2011)

Fundamenta Mathematicae

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We show that there is no uniformly continuous selection of the quotient map $Q : ℓ_{\infty} \to ℓ_{\infty} / c ₀$ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from $B_{X * *}$ onto $B_{X}$ .

On continuous composition operators

Wilhelmina Smajdor (2010)

Annales Polonici Mathematici

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Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that $M_{ψ} : = s u p | | [r, s, t; ψ] | | : r < s < t, r, s, t \in I < \infty$ , where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove...

Lipschitz spaces of holomorphic and pluriharmonic functions on bounded symmetric domains in $C^{N}$ (N>1)

Josephine Mitchell (1981)

Annales Polonici Mathematici

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$P^{λ}$ -spaces and $L^{λ}$ -spaces

Nguyen To Nhu (1979)

Colloquium Mathematicae

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